5. Hydrogen Atom

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The most important exactly solvable problem in quantum mechanics: explains atomic structure, spectroscopy, and the periodic table.

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Historical Context

1913

Bohr Model

Niels Bohr proposes quantized electron orbits explaining hydrogen spectrum

Significance: First successful quantum model of the atom, predicted Rydberg formula

1926

Schrödinger Solves Hydrogen

Erwin Schrödinger derives exact solution using his wave equation

Significance: Confirmed Bohr's energy levels, revealed true wave nature of electrons

1947

Lamb Shift Discovery

Willis Lamb measures 1057 MHz splitting between 2s and 2p levels

Significance: Led to development of Quantum Electrodynamics (QED)

💡 Understanding the historical development helps contextualize why certain concepts emerged and how they fit into the broader quantum revolution.

▶️

Video Lecture

Hydrogen Atom Solution - MIT OCW

Complete derivation of energy levels and orbitals by Barton Zwiebach

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

The Coulomb Potential

$$V(r) = -\frac{e^2}{4\pi\epsilon_0 r} = -\frac{ke^2}{r}$$

where $k = 1/(4\pi\epsilon_0) \approx 8.99 \times 10^9$ N·m²/C²

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Hydrogen Atom Energy Levels

Coulomb Potential

Selected Energy Level: n = 1

Number of Levels: 5

Show All Energy LevelsShow Wave Function

Legend:

Blue curve - Potential energy V(x)
Cyan line - Selected energy level
Green curve - Wave function ψ(x) at that energy
Gray dashed - Other energy levels

Radial Equation

For $u(r) = rR(r)$:

$$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + \left[-\frac{ke^2}{r} + \frac{\hbar^2\ell(\ell+1)}{2mr^2}\right]u = Eu$$

Derivation of Hydrogen Energy Levels

Assumptions:

Electron mass m, proton assumed infinitely massive
Non-relativistic: v << c
Pure Coulomb potential (no QED corrections)
Single electron (no electron-electron repulsion)

Starting with:

$$\left[-\frac{\hbar^2}{2m}\nabla^2 - \frac{ke^2}{r}\right]\psi = E\psi$$

Energy Levels

$$E_n = -\frac{me^4}{2(4\pi\epsilon_0)^2\hbar^2}\frac{1}{n^2} = -\frac{13.6\text{ eV}}{n^2}$$

where $n = 1, 2, 3, \ldots$ is the principal quantum number

Key result: Energy depends only on $n$, not on $\ell$ or $m$

Calculating Hydrogen Energy Levels

BASIC

Problem: Calculate the energy required to ionize a hydrogen atom from its ground state (n=1) and from the first excited state (n=2).

Given:

Ground state: n = 1
First excited state: n = 2
Energy formula: E_n = -13.6 eV / n²
Ionization means moving electron to E = 0

Find: Ionization energies from n=1 and n=2

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Hydrogen Spectral Lines Calculator

Calculate wavelengths of photons emitted during electron transitions

Formula:

$$\lambda = \frac{hc}{E_\gamma} = \frac{1240 \text{ eV·nm}}{|E_{n_f} - E_{n_i}|}$$

Initial State (nᵢ)

Final State (nf)

Notes:

Lyman series (nf=1): UV wavelengths
Balmer series (nf=2): Visible wavelengths (410-656 nm)
Paschen series (nf=3): Infrared wavelengths

Self-Check Question

Why does the hydrogen atom have degeneracy (multiple states with the same energy)?

Bohr Radius

Characteristic atomic size:

$$a_0 = \frac{4\pi\epsilon_0\hbar^2}{me^2} = 0.529 \text{ Å}$$

Ground state has peak probability density at $r = a_0$

▶️

Video Lecture

Quantum Angular Momentum - Stanford

Leonard Susskind explains quantum numbers and angular momentum commutation relations

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Quantum Numbers and Degeneracy

For each $n$:

$\ell = 0, 1, 2, \ldots, n-1$ ($n$ values)
For each $\ell$: $m = -\ell, \ldots, \ell$ ($2\ell + 1$ values)

Degeneracy:

$$g_n = \sum_{\ell=0}^{n-1}(2\ell+1) = n^2$$

Level $n$ has $n^2$ degenerate states (ignoring spin)

Calculating Degeneracy for n=3

BASIC

Problem: How many degenerate states does the n=3 energy level have in hydrogen? List all possible (ℓ, m) combinations.

Given:

Principal quantum number: n = 3
Allowed values: ℓ = 0, 1, ..., n-1
For each ℓ: m = -ℓ, -ℓ+1, ..., +ℓ

Find: Total number of states and all (ℓ, m) pairs

Self-Check Question

A hydrogen atom is in the n=4 level. How many different states (ignoring spin) have this same energy?

Radial Wave Functions

$$R_{n\ell}(r) = \sqrt{\left(\frac{2}{na_0}\right)^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]}}e^{-r/(na_0)}\left(\frac{2r}{na_0}\right)^\ell L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right)$$

where $L_q^p$ are associated Laguerre polynomials

First Few States

Ground state ($n=1, \ell=0, m=0$):

$$\psi_{100} = \frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}$$

First excited state ($n=2$):

$$\psi_{200} = \frac{1}{4\sqrt{2\pi a_0^3}}\left(2-\frac{r}{a_0}\right)e^{-r/(2a_0)}$$
$$\psi_{21m} = \frac{1}{8\sqrt{\pi a_0^3}}\frac{r}{a_0}e^{-r/(2a_0)}Y_1^m(\theta,\phi)$$

Spectroscopic Notation

States labeled as $n\ell$:

$n$	$\ell=0$	$\ell=1$	$\ell=2$	$\ell=3$
1	1s	—	—	—
2	2s	2p	—	—
3	3s	3p	3d	—
4	4s	4p	4d	4f

Self-Check Question

What does the spectroscopic notation '3d' represent?

Selection Rules

For electric dipole transitions:

$$\Delta\ell = \pm 1, \quad \Delta m = 0, \pm 1$$

No restriction on $\Delta n$

Radial Probability Distribution

Probability of finding electron between $r$ and $r + dr$:

$$P(r)dr = r^2|R_{n\ell}(r)|^2 dr$$

Factor $r^2$ from spherical volume element

▶️

Video Lecture

Visualizing Spherical Harmonics

eigenchris - Beautiful 3D visualizations of hydrogen orbitals

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Fine Structure

Small corrections lift degeneracy:

Relativistic correction: $\propto p^4$ term
Spin-orbit coupling: $\vec{L} \cdot \vec{S}$ interaction
Darwin term: From Dirac equation

Total fine structure splitting $\sim \alpha^2 E_n$ where $\alpha \approx 1/137$

Lamb Shift

QED correction splits $2s_{1/2}$ and $2p_{1/2}$ levels:

$$\Delta E_{\text{Lamb}} \approx 1057 \text{ MHz}$$

Due to vacuum fluctuations

Hydrogen-like Ions

For nuclear charge $Z$ (He⁺, Li²⁺, etc.):

$$E_n = -13.6Z^2/n^2 \text{ eV}$$

$$a = a_0/Z$$

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Real-World Applications of Hydrogen Atom Physics

1. Atomic Spectroscopy

Astronomy & Metrology

The hydrogen spectrum provides precise calibration for astronomical observations and laboratory measurements. The Balmer series lines are visible in stars and galaxies.

Examples:

Determining stellar compositions and velocities via Doppler shifts
Measuring redshifts of distant galaxies (Hubble expansion)
Atomic clocks using hydrogen maser transitions (10⁻¹⁵ accuracy)

2. Quantum Computing & Atomic Physics

Quantum Technology

Hydrogen-like atoms (ions) serve as qubits and test beds for quantum theory. Rydberg atoms with high n provide controllable quantum systems.

Examples:

Trapped ion qubits using Mg⁺, Ca⁺, Sr⁺ (hydrogen-like)
Rydberg atom quantum simulators
Precision tests of QED via Lamb shift measurements

3. Astrophysics & Cosmology

Cosmology

21 cm hyperfine line from ground state hydrogen maps the universe. Recombination era (z~1100) left cosmic microwave background.

Examples:

Radio astronomy: mapping galactic hydrogen clouds
Probing dark ages (first stars) via 21 cm cosmology
Understanding Big Bang nucleosynthesis and recombination

4. Chemistry & Molecular Modeling

Computational Chemistry

Hydrogen atom solutions form basis of quantum chemistry. Orbital shapes determine molecular bonding and reactivity.

Examples:

Density Functional Theory (DFT) for molecular simulations
Understanding chemical bonds (σ, π) from orbital overlap
Predicting molecular spectra and photochemistry

5. Fusion Energy Research

Energy

Understanding hydrogen and deuterium quantum states critical for magnetic and inertial confinement fusion.

Examples:

ITER tokamak design (hydrogen isotope plasmas)
Diagnostic spectroscopy in fusion reactors
Neutral beam injection energy calculations

💡 Understanding real-world applications helps connect abstract quantum concepts to tangible technology and motivates further study.

Summary

Key Equations

$$E_n = -\frac{13.6 \text{ eV}}{n^2}$$

$$a_0 = 0.529 \text{ Å}$$

$$g_n = n^2$$

$$\lambda = \frac{1240 \text{ eV·nm}}{E_{\gamma}}$$

Key Concepts

Coulomb potential leads to 1/n² energy scaling
Energy depends only on n (accidental degeneracy)
Each level has n² degenerate states (ℓ, m)
Bohr radius sets atomic size scale
Selection rules: Δℓ = ±1, Δm = 0, ±1
Fine structure and Lamb shift from QED
Basis for all atomic and molecular physics

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Runnable Simulations

Hydrogen Atom Radial Wave Functions and Probabilities

Python

script.py61 lines

import numpy as np
from math import factorial

# Hydrogen atom radial wave functions R_nl(r)
# R_nl(r) = sqrt((2/(n*a0))^3 * (n-l-1)!/(2n*((n+l)!)^3)) * exp(-r/(n*a0)) * (2r/(n*a0))^l * L_{n-l-1}^{2l+1}(2r/(n*a0))

a0 = 1.0  # Bohr radius in natural units

def laguerre(n, alpha, x):
    """Associated Laguerre polynomial L_n^alpha(x)"""
    if n == 0: return 1.0
    if n == 1: return 1.0 + alpha - x
    L_prev, L_curr = 1.0, 1.0 + alpha - x
    for k in range(2, n+1):
        L_next = ((2*k - 1 + alpha - x)*L_curr - (k - 1 + alpha)*L_prev) / k
        L_prev, L_curr = L_curr, L_next
    return L_curr

def R_nl(n, l, r):
    rho = 2*r/(n*a0)
    norm = np.sqrt((2/(n*a0))**3 * factorial(n-l-1)/(2*n*factorial(n+l)**3))
    # Corrected normalization
    norm = np.sqrt((2/(n*a0))**3 * factorial(n-l-1)/(2*n*(factorial(n+l))**1))
    return norm * np.exp(-rho/2) * rho**l * laguerre(n-l-1, 2*l+1, rho)

print("Hydrogen Atom: Energy Levels and Radial Probabilities")
print("E_n = -13.6 eV / n^2, <r>_nl = a0 * n^2 * (1 + (1/2)(1 - l(l+1)/n^2))")
print("=" * 60)

# Energy levels
for n in range(1, 6):
    E = -13.6 / n**2
    degeneracy = 2 * n**2
    print(f"n = {n}: E = {E:8.4f} eV, degeneracy = {degeneracy}")

# Expectation values
print(f"\n{'(n,l)':>8s}  {'<r>/a0':>10s}  {'<r^2>/a0^2':>12s}  {'r_max/a0':>10s}")
print("-" * 45)

for n in range(1, 5):
    for l in range(n):
        r_avg = n**2 * a0 * (1 + 0.5*(1 - l*(l+1)/n**2))
        r2_avg = a0**2 * n**2 * (n**2 + 2) / 2 * (1 + 0) # simplified
        r_max = a0 * n**2  # most probable r for l=n-1
        if l == n-1:
            r_max = n**2 * a0
        else:
            r_max = n * a0 * (n - 0.5)  # approximate
        print(f"({n},{l}):  {r_avg:10.4f}  {r2_avg:12.2f}  {r_max:10.4f}")

# Radial probability density at specific points
print(f"\nRadial probability P(r) = r^2 |R_nl(r)|^2 for (1,0) and (2,0):")
print(f"{'r/a0':>8s}  {'P_10(r)':>10s}  {'P_20(r)':>10s}  {'P_21(r)':>10s}")
print("-" * 42)
for r in np.linspace(0.1, 12, 12):
    P10 = r**2 * (2*np.exp(-r))**2
    P20 = r**2 * (1/(2*np.sqrt(2)) * (2-r) * np.exp(-r/2))**2
    P21 = r**2 * (1/(2*np.sqrt(6)) * r * np.exp(-r/2))**2
    print(f"{r:8.2f}  {P10:10.6f}  {P20:10.6f}  {P21:10.6f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Hydrogen Spectral Series: Lyman, Balmer, Paschen

Fortran

program.f9076 lines

program hydrogen_spectrum
  implicit none
  double precision, parameter :: eV_to_nm = 1239.84d0
  double precision :: E_n, E_m, delta_E, wavelength
  double precision :: R_inf
  integer :: n, m, count

R_inf = 13.6d0  ! Rydberg energy in eV

print *, "Hydrogen Atom Spectral Series"
  print *, "1/lambda = R_H * (1/n^2 - 1/m^2)"
  print *, "======================================"
  print *, ""

! Lyman series (n=1)
  print *, "--- Lyman Series (n=1, UV) ---"
  print '(A6, A12, A14, A14)', " m", "E (eV)", "lambda (nm)", "region"
  print *, "--------------------------------------------"
  n = 1
  do m = 2, 7
    E_n = -R_inf / dble(n)**2
    E_m = -R_inf / dble(m)**2
    delta_E = E_m - E_n   ! negative (absorption) -> use |delta_E|
    delta_E = abs(delta_E)
    wavelength = eV_to_nm / delta_E
    print '(I6, F12.4, F14.2, A14)', m, delta_E, wavelength, "  UV"
  end do

! Balmer series (n=2)
  print *, ""
  print *, "--- Balmer Series (n=2, Visible/UV) ---"
  print '(A6, A12, A14, A14)', " m", "E (eV)", "lambda (nm)", "color"
  print *, "--------------------------------------------"
  n = 2
  do m = 3, 8
    E_n = -R_inf / dble(n)**2
    E_m = -R_inf / dble(m)**2
    delta_E = abs(E_m - E_n)
    wavelength = eV_to_nm / delta_E
    if (wavelength > 620) then
      print '(I6, F12.4, F14.2, A14)', m, delta_E, wavelength, "  red"
    else if (wavelength > 495) then
      print '(I6, F12.4, F14.2, A14)', m, delta_E, wavelength, "  cyan"
    else if (wavelength > 450) then
      print '(I6, F12.4, F14.2, A14)', m, delta_E, wavelength, "  blue"
    else
      print '(I6, F12.4, F14.2, A14)', m, delta_E, wavelength, "  violet"
    end if
  end do

! Paschen series (n=3)
  print *, ""
  print *, "--- Paschen Series (n=3, IR) ---"
  print '(A6, A12, A14, A14)', " m", "E (eV)", "lambda (nm)", "region"
  print *, "--------------------------------------------"
  n = 3
  do m = 4, 9
    E_n = -R_inf / dble(n)**2
    E_m = -R_inf / dble(m)**2
    delta_E = abs(E_m - E_n)
    wavelength = eV_to_nm / delta_E
    print '(I6, F12.4, F14.2, A14)', m, delta_E, wavelength, "  IR"
  end do

! Series limits
  print *, ""
  print *, "Series limits (m -> infinity):"
  do n = 1, 4
    wavelength = eV_to_nm / (R_inf / dble(n)**2)
    print '(A, I2, A, F10.2, A)', &
      "  n=", n, ": lambda_min = ", wavelength, " nm"
  end do

end program hydrogen_spectrum

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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5. Hydrogen Atom

Your Progress

Historical Context

Bohr Model

Schrödinger Solves Hydrogen

Lamb Shift Discovery

Video Lecture

The Coulomb Potential

Hydrogen Atom Energy Levels

Radial Equation

Derivation of Hydrogen Energy Levels

Energy Levels

Calculating Hydrogen Energy Levels

Hydrogen Spectral Lines Calculator

Self-Check Question

Bohr Radius

Video Lecture

Quantum Numbers and Degeneracy

Calculating Degeneracy for n=3

Self-Check Question

Radial Wave Functions

First Few States

Spectroscopic Notation

Self-Check Question

Selection Rules

Radial Probability Distribution

Video Lecture

Fine Structure

Lamb Shift

Hydrogen-like Ions

Real-World Applications of Hydrogen Atom Physics

1. Atomic Spectroscopy

2. Quantum Computing & Atomic Physics

3. Astrophysics & Cosmology

4. Chemistry & Molecular Modeling

5. Fusion Energy Research

Summary

Key Equations

Key Concepts

Related Topics

Spherical Harmonics

Angular Momentum

3D Harmonic Oscillator

Spin

Fine Structure

Stark Effect

Zeeman Effect

Multi-Electron Atoms

Runnable Simulations

Hydrogen Atom Radial Wave Functions and Probabilities

Hydrogen Spectral Series: Lyman, Balmer, Paschen